On the exact electric and magnetic ﬁelds of an electric dipole

W.J.M.Kort-Kamp

a

and C.Farina

b

Instituto de Fisica,Universidade Federal do Rio de Janeiro,Rio de Janeiro 21945-970,Brazil

Received 3 May 2010;accepted 24 August 2010

We make a multipole expansion directly in Jeﬁmenko’s equations to obtain the exact expressions for

the electric and magnetic ﬁelds of an electric dipole with an arbitrary time dependence.Some

comments are made about the usual derivations in most undergraduate and graduate textbooks and

in literature.©

2011 American Association of Physics Teachers.

DOI:10.1119/1.3488989

The problemof ﬁnding an analytic expression for the elec-

tric ﬁeld of a localized but arbitrary static charge distribution

is quite involved.Due to the difﬁculty in obtaining exact

solutions,numerical methods and approximate theoretical

methods have been developed.One of the most important

examples of the latter is the multipole expansion method.For

the origin inside the distribution,the multipole expansion

method gives the ﬁeld outside the distribution as a superpo-

sition of ﬁelds,each of which can be interpreted as the elec-

trostatic ﬁeld of a multipole located at the origin see,for

instance,Ref.1.The ﬁrst three terms of the multipole ex-

pansion correspond to the ﬁelds of a monopole,a dipole,and

a quadrupole,respectively.For an arbitrary distribution with

a vanishing total charge,the ﬁrst term in the multipole ex-

pansion is the ﬁeld of the electric dipole of the distribution.

This property is an example of the important role played by

the ﬁeld of an electric dipole.

For localized charge and current distributions with arbi-

trary time dependence,the multipole expansion method is a

powerful technique for calculating the electromagnetic ﬁelds

of the system.In the study of radiation ﬁelds of simple sys-

tems,such as antennas,it is common to make a multipole

expansion and to keep only the ﬁrst few terms.It is worth

emphasizing that in contrast to what happens in the case of

static charge distributions,the leading contribution to the ra-

diation ﬁelds comes from the electric dipole term since there

is no monopole radiation due to charge conservation.

The electric ﬁeld of an electric dipole has three

contributions—the static zone contribution proportional to

1/r

3

,the intermediate zone contribution proportional to 1/r

2

,

and the far zone or radiation contribution proportional to 1/r,

where r is the distance from the origin.As we shall see,the

corresponding magnetic ﬁeld has only two contributions. A

general derivation of the exact electric and magnetic ﬁelds of

an electric dipole is lacking from most textbooks.

2–12

In

some cases,the textbooks obtain the exact electromagnetic

ﬁelds for the electric dipole by assuming a harmonic time

dependence for the sources

2–5

or by assuming a particular

model for the charge and current distributions.

6–8

In other

cases,the textbooks are interested only in the radiation

ﬁelds.

9–12

A general derivation of the exact electric and magnetic

ﬁelds of an electric dipole with arbitrary time dependence

can be found in Ref.13.Heras ﬁrst presented a new version

of Jeﬁmenko’s equations in a material medium

14

with polar-

ization P and magnetization M and then obtained the exact

dipole ﬁelds.

The purpose of this note is to provide a direct derivation of

the exact electric and magnetic ﬁelds of an electric dipole

with arbitrary time dependence without making any assump-

tions for the localized charge and current distributions of

the system.Our starting point will be Jeﬁmenko’s equations

and our procedure consists of making a multipole expansion

up to the desired order.Our procedure does not require Jeﬁ-

menko’s equation in matter,which makes our derivation a

very simple one.

Jeﬁmenko’s equations for arbitrary but localized sources

are given by

15–19

Er,t =

1

4

0

dr

r

,t

R

R

3

+

dr

˙

r

,t

R

cR

2

−

dr

J

˙

r

,t

c

2

R

,1

Br,t =

0

4

dr

Jr

,t

R

R

3

+

dr

J

˙

r

,t

R

cR

2

,2

where the integrals are over all space,R=R =r−r

,the

overdot means time derivative for example,

˙

r

,t

=r

,t

/t

,and the brackets … mean that the quanti-

ties inside them must be evaluated at the retarded time t

=t −R/c.

We make a multipole expansion,which consists of making

an expansion in powers of r

/r,which is valid outside the

charge and current distributions.Our ﬁnal result will be valid

only outside the sources.Because we are interested in the

dipole ﬁelds,we shall retain only terms up to linear order in

r

.Therefore,we use the following Taylor expansions:

1

R

1

r

+

r

ˆ

r

r

2

,3

R

R

2

r

ˆ

r

+

2r

ˆ

r

r

ˆ

r

2

−

r

r

2

,4

R

R

3

r

ˆ

r

2

+

3r

ˆ

r

r

ˆ

r

3

−

r

r

3

,5

111 111Am.J.Phys.79 1,January 2011 http://aapt.org/ajp © 2011 American Association of Physics Teachers

t −

R

c

t

0

+

r

ˆ

r

c

,6

where t

0

=t −r/c is the retarded time with respect to the ori-

gin.Analogously,we expand the source terms about t

=t

0

and keep terms only up to linear order in r

.The results are

r

,t

=r

,t − R/c

r

,t

0

+ r

ˆ

r

/c r

,t

0

+

r

ˆ

r

c

˙

r

,t

0

,

7

Jr

,t

= Jr

,t − R/c

Jr

,t

0

+ r

ˆ

r

/c Jr

,t

0

+

r

ˆ

r

c

J

˙

r

,t

0

,

8

where we used Eq.6.We can make analogous expansions

to obtain approximate expressions for the time derivatives of

the sources,namely,

˙

r

,t

˙

r

,t

0

+

r

ˆ

r

c

¨

r

,t

0

,9

J

˙

r

,t

J

˙

r

,t

0

+

r

ˆ

r

c

J

¨

r

,t

0

.10

For convenience,we denote the three integrals that appear on

the right-hand side of Eq.1 by I

1

E

,I

2

E

,and I

3

E

.Similarly,

we denote by I

1

B

and I

2

B

the two integrals that appear on the

right-hand side of Eq.2.

We must calculate these integrals up to the desired order.

We ﬁrst consider I

1

E

,substitute Eqs.5 and 7 into the

expression for I

1

E

,and obtain

I

1

E

dr

r

,t

0

+

r

ˆ

r

c

˙

r

,t

0

r

ˆ

r

2

+

3r

ˆ

r

r

ˆ

r

3

−

r

r

3

11

r

ˆ

r

2

dr

r

,t

0

+

3r

ˆ

r

3

r

ˆ

dr

r

,t

0

r

−

1

r

3

dr

r

,t

0

r

+

r

ˆ

cr

2

r

ˆ

dr

˙

r

,t

0

r

,

12

where quadratic terms in r

are neglected.The integral in the

ﬁrst termon the right-hand side of Eq.12 is the total charge

of the distribution,which is independent of time due to

charge conservation.This term corresponds to the monopole

termof the multipole expansion because it is the ﬁeld created

by a point charge Qr

,t

0

dr

ﬁxed at the origin.This

term does not contribute to the dipole ﬁeld.We can identify

the electric dipole moment of the distribution at time t

0

in the

next two integrals on the right-hand side of Eq.12,namely,

pt

0

=dr

r

,t

0

r

.After we interchange the time deriva-

tive with the integration in the last integral on the right-hand

side of Eq.12,we can identify the time derivative of the

electric dipole moment of the distribution at time t

0

,namely,

p

˙

t

0

=d/dtdr

r

,t

0

r

.Hence,I

1

E

is given by

I

1

E

Qr

ˆ

r

2

+

3r

ˆ

pt

0

r

ˆ

− pt

0

r

3

+

r

ˆ

p

˙

t

0

cr

2

.13

We next use an analogous procedure and substitute into

the expression for I

2

E

the approximations in Eqs.4 and 9.

The result is

I

2

E

2r

ˆ

p

˙

t

0

r

ˆ

− p

˙

t

0

cr

2

+

r

ˆ

p

¨

t

0

r

ˆ

c

2

r

.14

To calculate I

3

E

,I

1

B

,and I

2

B

,it is convenient to write the

Cartesian basis vectors as e

ˆ

i

=

x

i

i =1,2,3 so that any

vector b can be written in the form b=b

i

e

ˆ

i

=b e

ˆ

i

e

ˆ

i

=b

x

i

e

ˆ

i

,where the Einstein convention of summation

over repeated indices is assumed.As it will become clear,it

sufﬁces to make the approximations R→r and J

˙

r

,t

→J

˙

r

,t

0

in the integrand.Hence,we have

I

3

E

−

1

c

2

r

dr

J

˙

r

,t

0

= −

1

c

2

r

e

ˆ

i

dr

ˆJ

˙

r

,t

0

x

i

‰ 15a

=

1

c

2

r

e

ˆ

i

dr

x

i

ˆ

J

˙

r

,t

0

‰

= −

1

c

2

r

e

ˆ

i

dr

x

i

¨

r

,t

0

,15b

where in passing from Eq.15a to Eq.15b we integrated

by parts and discarded the surface term because the integra-

tion is over all space and the sources are localized.In the last

step we used the continuity equation.The second time de-

rivative of the electric dipole moment of the distribution at

time t

0

appears in Eq.14,and thus we obtain

I

3

E

−

p

¨

t

0

c

2

r

.16

We now see why it was not necessary to go beyond zeroth

order in the expansion of J

˙

r

,t

in the calculation of I

3

E

.

The electric dipole moment of the distribution is already ﬁrst

order in r

so that the next order termwould contribute to the

next terms of the multipole expansion,namely,the magnetic

dipole term and the electric quadrupole term.

The integrals I

1

B

and I

2

B

will give the contribution to the

magnetic ﬁeld of the electric dipole term.We proceed as we

did for I

3

E

,namely,it sufﬁces to use only the zeroth order

approximation for the expansions of the current and its time

derivative.We have

112 112Am.J.Phys.,Vol.79,No.1,January 2011 W.J.M.Kort-Kamp and C.Farina

I

1

B

−

r

ˆ

r

2

dr

Jr

,t

0

17a

=−

r

ˆ

r

2

e

ˆ

i

dr

ˆJr

,t

0

x

i

‰ 17b

=

r

ˆ

r

2

e

ˆ

i

dr

x

i

ˆ

Jr

,t

0

‰ 17c

=−

r

ˆ

r

2

e

ˆ

i

dr

x

i

˙

r

,t

0

=

p

˙

t

0

r

ˆ

r

2

,17d

where in the last step we identiﬁed the time derivative of the

electric dipole moment of the distribution.As before,it is

straightforward to show that

I

2

B

p

¨

t

0

r

ˆ

cr

.18

We substitute the results in Eqs.13,14,and 16 into

Eq.1,discard the monopole term,substitute Eqs.17d and

18 into Eq.2,and obtain the exact electric and magnetic

ﬁelds associated with the electric dipole term of an arbitrary

localized distribution of charges and currents,

Er,t =

1

4

0

3r

ˆ

pt

0

r

ˆ

− pt

0

r

3

+

3r

ˆ

p

˙

t

0

r

ˆ

− p

˙

t

0

cr

2

+

r

ˆ

r

ˆ

p

¨

t

0

c

2

r

,19

Br,t =

0

4

p

˙

t

0

r

ˆ

r

2

+

p

¨

t

0

r

ˆ

cr

,20

which are the desired ﬁelds.Equation 19 is given,but not

derived,in Ref.20.

The last terms on the right-hand side of Eqs.19 and 20

are proportional to 1/r and correspond to the radiation ﬁelds

of an electric dipole.For the idealized case of a point electric

dipole ﬁxed at the origin,Eqs.19 and 20 are exact for

every point in space,except the origin,where the ﬁelds are

singular.Because we are interested in the ﬁelds outside the

dipole,we shall not be concerned with these singular terms.

A discussion of these terms for a static point electric dipole

can be found in Ref.2,Chap.3.

As an important special case,we consider an electric di-

pole with harmonic time dependence.We substitute pt

=p

0

e

−it

into Eqs.19 and 20 and obtain the well known

results,

2

Er,t =

e

−it

4

0

k

2

r

ˆ

p

0

r

ˆ

e

ikr

r

+ 3r

ˆ

p

0

r

ˆ

− p

0

1

r

3

−

ik

r

2

e

ikr

,21

Br,t =

0

e

−it

4

ck

2

r

ˆ

p

0

e

ikr

r

1 −

1

ikr

,22

where k=/c.In Eq.19 note the presence of a term pro-

portional to r

−3

and pt

0

,which is a characteristic of the

ﬁeld of a static electric dipole,except that here the electric

dipole moment is evaluated at the retarded time t

0

.This term

dominates in the near zone,where dr,where d is a

typical length scale of the source and is the wavelength of

the electromagnetic ﬁeld.Note the absence of such a term in

Eq.20,which is expected because a static electric dipole

does not create a magnetic ﬁeld.In the radiation zone,where

dr,the dominant terms are proportional to r

−1

and to

the second time derivative of the electric dipole moment.The

transverse character of the radiation ﬁelds is evident.

The results in Eqs.19 and 20 could have been obtained

if,instead of Jeﬁmenko’s Eq.1 for the electric ﬁeld,we had

used the equivalent Panofsky–Phillips equation for the elec-

tric ﬁeld

11

given by

Er,t =

1

4

0

dr

r

,t

R

ˆ

R

2

+

dr

Jr

,t

R

ˆ

R

ˆ

+ Jr

,t

R

ˆ

R

ˆ

cR

2

+

1

4

0

dr

J

˙

r

,t

R

ˆ

R

ˆ

c

2

R

.23

Equation 23 is equivalent to Eq.1,as it was shown ex-

plicitly in Ref.21,but note that Eq.23 is more convenient

as a starting point for calculating multipole radiation ﬁelds.

22

All multipole radiation ﬁelds can be obtained from the last

terms of the right-hand side of Eqs.2 and 23.The dipole

radiation ﬁelds in Eqs.19 and 20 are just the ﬁrst order

contributions of the multipole expansion for the radiation

ﬁelds.The next order contributions,namely,the radiation

ﬁelds of a magnetic dipole and an electric quadrupole,may

be similarly obtained without much more effort.

22

We leave for the interested reader the problem of obtain-

ing the exact electric ﬁeld of an electric dipole with arbitrary

time dependence from Eq.23 instead of Eq.1.The exact

electric and magnetic ﬁelds of higher multipoles such as the

magnetic dipole and electric quadrupole can also be obtained

following our approach.In this case,higher order terms in r

must be kept.

We have calculated the exact electric and magnetic ﬁelds

of an electric dipole with arbitrary time dependence,in con-

trast to the usual calculations where a harmonic time depen-

dence is assumed.Most textbooks use the electromagnetic

potentials.We have instead implemented a multipole expan-

sion directly from Jeﬁmenko’s equations.Besides the sim-

plicity of our method,it increases the number of problems

that can be attacked directly by using Jeﬁmenko’s equations

see also Refs.23–28.

113 113Am.J.Phys.,Vol.79,No.1,January 2011 W.J.M.Kort-Kamp and C.Farina

ACKNOWLEDGMENTS

The authors are indebted to Professor C.Sigaud for read-

ing the paper.They also would like to thank CNPq for partial

ﬁnancial support.

a

Electronic mail:kortkamp@if.ufrj.br

b

Electronic mail:farina@if.ufrj.br

1

David J.Grifﬁths,Introduction to Electrodynamics,3rd ed.Prentice-

Hall,Englewood Cliffs,NJ,1999,Chap.3.

2

J.D.Jackson,Classical Electrodynamics,3rd ed.Wiley,New York,

1999,Chap.9.

3

Emil J.Konopinsky,Electromagnetic Fields and Relativistic Particles

McGraw-Hill,New York,1981,Chap.8.

4

William R.Smythe,Static and Dynamic Electricity McGraw-Hill,New

York,1939,Chap.8.

5

J.A.Stratton,Electromagnetic Theory McGraw-Hill,New York,1941,

Chap.8.

6

R.P.Feynman,R.B.Leighton,and M.Sands,The Feynman Lectures on

Physics Addison-Wesley,Reading,MA,1964,Vol.2,Chap.21.

7

A.Sommerfeld,Electrodynamics Academic,New York,1952,Vol.3,

Sec.19.

8

Mark A.Heald and Jerry B.Marion,Classical Electromagnetic Radia-

tion,3rd ed.Saunders College Publishing,New York,1995,Chap.9.

9

L.D.Landau and E.M.Lifshitz,The Classical Theory of Fields

Addison-Wesley,Reading,MA,1962,Chap.9.

10

Reference 1,Chap.11.

11

W.K.H.Panofsky and M.Phillips,Classical Electricity and Magnetism,

2nd ed.Addison-Wesley,Reading,MA,1962,Chap.14.

12

John R.Reitz and Frederick J.Milford,Foundations of Electromagnetic

Theory Addson-Wesley,Reading,MA,1962,Chap.16.

13

J.A.Heras,“Radiation ﬁelds of a dipole in arbitrary motion,” Am.J.

Phys.62,1109–1115 1994.

14

Oleg D.Jeﬁmenko,“Solutions of Maxwell’s equations for electric and

magnetic ﬁelds in arbitrary media,” Am.J.Phys.60,899–902 1992.

15

Oleg D.Jeﬁmenko,Electricity and Magnetism Appleton-Century-Crofts,

New York,1966,Chap.15.

16

J.A.Heras,“Time-dependent generalizations of the Biot-Savart and Cou-

lomb laws:A formal derivation,” Am.J.Phys.63,928–932 1995.

17

D.J.Grifﬁths and M.A.Heald,“Time-dependent generalizations of the

Biot-Savart and Coulomb law,” Am.J.Phys.59,111–117 1991.

18

Tran-Cong Ton,“On the time-dependent,generalized Coulomb and Biot-

Savart laws,” Am.J.Phys.59,520–528 1991.

19

U.Bellotti and M.Bornatici,“Time-dependent,generalized Coulomb and

Biot-Savart laws:A derivation based on Fourier transforms,” Am.J.

Phys.64,568–570 1996.

20

Peter W.Milonni,The Quantum Vacuum:An Introduction to Electrody-

namics Academic,San Diego,1994.

21

K.T.McDonald,“The relation between expressions for time-dependent

electromagnetic ﬁelds given by Jeﬁmenko and by Panofsky and Phillips,”

Am.J.Phys.65,1074–1076 1997.

22

R.de Melo e Souza,M.V.Cougo-Pinto,C.Farina,and M.Moriconi,

“Multipole radiation ﬁelds from the Jeﬁmenko equation for the magnetic

ﬁeld and the Panosfsky-Phillips equation for the electric ﬁeld,” Am.J.

Phys.77,67–72 2009.

23

J.A.Heras,“Can Maxwell’s equations be obtained from the continuity

equation?,” Am.J.Phys.75,652–657 2007.

24

J.A.Heras,“Jeﬁmenko’s formulas with magnetic monopoles and the

Lienard-Wiechert ﬁelds of a dual-charged particle,” Am.J.Phys.62,

525–531 1994.

25

F.Rohrlich,“Causality,the Coulomb ﬁeld,and Newton’s law of gravita-

tion,” Am.J.Phys.70,411–414 2002.

26

Oleg D.Jeﬁmenko,“Comment on ‘Causality,the Coulomb ﬁeld,and

Newton’s law of gravitation’,” Am.J.Phys.70,964–964 2002.

27

F.Rohrlich,“Reply to ‘Comment on “causality,the Coulomb ﬁeld,and

Newton’s law of gravitation’,” Am.J.Phys.70,964–964 2002.

28

J.A.Heras,“New approach to the classical radiation ﬁelds of moving

dipoles,” Phys.Lett.A 237,343–348 1998.

114 114Am.J.Phys.,Vol.79,No.1,January 2011 W.J.M.Kort-Kamp and C.Farina

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